Finance and Stochastics

, Volume 19, Issue 4, pp 763–790 | Cite as

Aggregation-robustness and model uncertainty of regulatory risk measures

Article

Abstract

Research related to aggregation, robustness and model uncertainty of regulatory risk measures, for instance, value-at-risk (VaR) and expected shortfall (ES), is of fundamental importance within quantitative risk management. In risk aggregation, marginal risks and their dependence structure are often modelled separately, leading to uncertainty arising at the level of a joint model. In this paper, we introduce a notion of qualitative robustness for risk measures, concerning the sensitivity of a risk measure to the uncertainty of dependence in risk aggregation. It turns out that coherent risk measures, such as ES, are more robust than VaR according to the new notion of robustness. We also give approximations and inequalities for aggregation and diversification of VaR under dependence uncertainty, and derive an asymptotic equivalence for worst-case VaR and ES under general conditions. We obtain that for a portfolio of a large number of risks, VaR generally has a larger uncertainty spread compared to ES. The results warn that unjustified diversification arguments for VaR used in risk management need to be taken with much care, and they potentially support the use of ES in risk aggregation. This in particular reflects on the discussions in the recent consultative documents by the Basel Committee on Banking Supervision.

Keywords

Value-at-risk Expected shortfall Dependence uncertainty Risk aggregation Aggregation-robustness Inhomogeneous portfolio Basel III 

Mathematics Subject Classification

62G35 60E15 62P05 

JEL Classification

C10 

References

  1. 1.
    Aas, K., Puccetti, G.: Bounds for total economic capital: the DNB case study. Extremes 17(4), 693–715 (2014) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Acerbi, C.: Spectral measures of risk: a coherent representation of subjective risk aversion. J. Bank. Finance 26, 1505–1518 (2002) CrossRefGoogle Scholar
  3. 3.
    Artzner, P., Delbaen, F., Eber, J.-M., Heath, D.: Coherent measures of risk. Math. Finance 9, 203–228 (1999) MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    BCBS: Consultative Document May 2012. Fundamental review of the trading book. Basel Committee on Banking Supervision. Bank for International Settlements, Basel (2012). Available online http://www.bis.org/publ/bcbs219.htm
  5. 5.
    BCBS: Consultative Document October 2013. Fundamental review of the trading book: a revised market risk framework. Basel Committee on Banking Supervision. Bank for International Settlements, Basel (2013). Available online http://www.bis.org/publ/bcbs265.htm
  6. 6.
    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009) CrossRefMATHGoogle Scholar
  7. 7.
    Bernard, C., Jiang, X., Wang, R.: Risk aggregation with dependence uncertainty. Insur. Math. Econ. 54, 93–108 (2014) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cambou, M., Filipović, D.: Model uncertainty and scenario aggregation. Math. Finance (2015, to appear). doi:10.1111/mafi.12097 Google Scholar
  9. 9.
    Cheung, K.C., Lo, A.: General lower bounds on convex functionals of aggregate sums. Insur. Math. Econ. 53, 884–896 (2013) MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cont, R., Deguest, R., Scandolo, G.: Robustness and sensitivity analysis of risk measurement procedures. Quant. Finance 10, 593–606 (2010) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Daníelsson, J., Jorgensen, B.N., Mandira, S., Samorodnitsky, G., de Vries, C.G.: Subadditivity re-examined: the case for Value-at-Risk. Discussion paper, 549. Financial Markets Group, London School of Economics and Political Science. Available online http://eprints.lse.ac.uk/24668
  12. 12.
    EIOPA: Equivalence assessment of the Swiss supervisory system in relation to articles 172, 227 and 260 of the Solvency II Directive, EIOPA-BoS-11-028 (2011). Available online https://eiopa.europa.eu/Pages/SearchResults.aspx?k=filename:EIOPA-BoS-11-028-Swiss-Equivalence-advice.pdf
  13. 13.
    Embrechts, P., Puccetti, G.: Bounds for functions of dependent risks. Finance Stoch. 10, 341–352 (2006) MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Embrechts, P., Puccetti, G., Rüschendorf, L.: Model uncertainty and VaR aggregation. J. Bank. Finance 37, 2750–2764 (2013) CrossRefGoogle Scholar
  15. 15.
    Embrechts, P., Puccetti, G., Rüschendorf, L., Wang, R., Beleraj, A.: An academic response to Basel 3.5. Risks 2, 25–48 (2014) CrossRefGoogle Scholar
  16. 16.
    Emmer, S., Kratz, M., Tasche, D.: What is the best risk measure in practice? A comparison of standard measures. Preprint, ESSEC Business School (2014). Available online http://ssrn.com/abstract=2370378
  17. 17.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. de Gruyter, Berlin (2011) CrossRefMATHGoogle Scholar
  18. 18.
    Gneiting, T.: Making and evaluating point forecasts. J. Am. Stat. Assoc. 106, 746–762 (2011) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Hampel, F.: A general qualitative definition of robustness. Ann. Math. Stat. 42, 1887–1896 (1971) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Hansen, L.P., Sargent, T.J.: Robustness. Princeton University Press, Princeton (2007) MATHGoogle Scholar
  21. 21.
    Huber, P.J., Ronchetti, E.M.: Robust Statistics, 2nd edn. Wiley Series in Probability and Statistics. Wiley, New York (2009) CrossRefMATHGoogle Scholar
  22. 22.
    Jorion, P.: Value at Risk: The New Benchmark for Managing Financial Risk, 3rd edn. McGraw-Hill, New York (2006) Google Scholar
  23. 23.
    Kou, S., Peng, X.: On the measurement of economic tail risk. Preprint (2014). Available online arXiv:1401.4787
  24. 24.
    Krätschmer, V., Schied, A., Zähle, H.: Qualitative and infinitesimal robustness of tail-dependent statistical functionals. J. Multivar. Anal. 103, 35–47 (2012) MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Krätschmer, V., Schied, A., Zähle, H.: Comparative and quantitative robustness for law-invariant risk measures. Finance Stoch. 18, 271–295 (2014) MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Kusuoka, S.: On law invariant coherent risk measures. Adv. Math. Econ. 3, 83–95 (2001) MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Makarov, G.D.: Estimates for the distribution function of the sum of two random variables with given marginal distributions. Theory Probab. Appl. 26, 803–806 (1981) MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nelsen, R.: An Introduction to Copulas, 2nd edn. Springer, New York (2006) MATHGoogle Scholar
  29. 29.
    Puccetti, G.: Sharp bounds on the expected shortfall for a sum of dependent random variables. Stat. Probab. Lett. 83, 1227–1232 (2013) MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Puccetti, G., Rüschendorf, L.: Sharp bounds for sums of dependent risks. J. Appl. Probab. 50, 42–53 (2013) MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Puccetti, G., Rüschendorf, L.: Asymptotic equivalence of conservative value-at-risk- and expected shortfall-based capital charges. J. Risk 16(3), 1–19 (2014) Google Scholar
  32. 32.
    Puccetti, G., Wang, B., Wang, R.: Complete mixability and asymptotic equivalence of worst-possible VaR and ES estimates. Insur. Math. Econ. 53, 821–828 (2013) MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Rüschendorf, L.: Random variables with maximum sums. Adv. Appl. Probab. 14, 623–632 (1982) MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Sandström, A.: Handbook of Solvency for Actuaries and Risk Managers: Theory and Practice. Taylor & Francis, Florida (2010) CrossRefMATHGoogle Scholar
  35. 35.
    SCOR: From Principle-Based Risk Management to Solvency Requirements, 2nd edn. SCOR, Zurich (2008). SCOR Switzerland AG. Available online http://www.scor.com/images/stories/pdf/scorpapers/sstbook_second_edition_final.pdf Google Scholar
  36. 36.
    van der Vaart, A.W.: Asymptotic Statistics. Cambridge University Press, Cambridge (1998) CrossRefMATHGoogle Scholar
  37. 37.
    Wang, B., Wang, R.: The complete mixability and convex minimization problems for monotone marginal distributions. J. Multivar. Anal. 102, 1344–1360 (2011) CrossRefMATHGoogle Scholar
  38. 38.
    Wang, B., Wang, R.: Extreme negative dependence and risk aggregation. J. Multivar. Anal. 136, 12–25 (2015). MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    Wang, R.: Asymptotic bounds for the distribution of the sum of dependent random variables. J. Appl. Probab. 51, 780–798 (2014) MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Wang, R., Peng, L., Yang, J.: Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities. Finance Stoch. 17, 395–417 (2013) MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Wang, S.S., Young, V.R., Panjer, H.H.: Axiomatic characterization of insurance prices. Insur. Math. Econ. 21, 173–183 (1997) MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of MathematicsRiskLab and SFI, ETH ZurichZurichSwitzerland
  2. 2.Department of MathematicsBeijing Technology and Business UniversityBeijingChina
  3. 3.Department of Statistics and Actuarial ScienceUniversity of WaterlooWaterlooCanada

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